Presentation Details 

Avital, Eldad 
Sound Scattering and Control by Free Surface Piercing and Fluid Loaded Cylindrical Shells 
View Abstract
Hide Abstract

The interaction between a monochromic sound wave and a thin cylindrical shell piercing shallow water surface is considered. The cylindrical shell is clamped to the floor and may be filled with fluid or is empty inside. This model problem is relevant for a variety of applications ranging from offshore structures to underwater vessels.
Linear acoustics and elastic shell dynamics are assumed. Fourier transforms in the vertical and azimuthal directions are used and the shell’s deflection is found by matching its acceleration with the outer and inner fluid pressures. Sound control is achieved by assumed a distribution of pressure actuators acting on the cylindrical wall.
Sound field analysis is carried out for an Aluminium shell of 30 cm diameter and 5 meter depth and for low to mid range sound frequencies of 5000 Hz to 20KHz. It is shown that adding fluid inside the cylindrical shell can reduce the scattered sound field level, but does not profoundly change it, still showing reflections in front and sidewise of the shell and a V wake behind it. An equation for a continuous control pressure to cancel the scattered sound is derived and the effect of using a practical configuration of discrete pressure line actuators is shown to have a moderate to a good capability of reducing the scattered sound wave. 
Bennetts, Luke 
Numerical and experimental modelling of synthetic `sea ice' in a wavetank 
View Abstract
Hide Abstract

Sea ice floes that form in the polar oceans are modelled as floating thinelastic plates, as their horizontal dimensions far exceed their thicknesses and as they are known to flex in response to wave forcing. Much theoretical work has been achieved in modelling the interaction of ocean waves with regions of sea ice over the last twenty years, especially under linear and time harmonic assumptions. However, the same period has been devoid of experimental data, which hinders progress. Whilst field work in this area involves long, expensive expeditions in unpredictable conditions, it is hoped that some insight can be gained from wavetank experiments on synthetic floes that will be conducted this year. An outline of these will be presented along with complementary mathematical models. 
Blyth, Mark 
Transit of an elastic capsule through a branching channel 
View Abstract
Hide Abstract

The transit of an elastic fluidfilled capsule through a channel with a side branch is investigated numerically using the boundary integral method for Stokes flow.
The capsule is carried in a pressuredriven flow of fluid of generally different viscosity to that inside the capsule. Far upstream and downstream in the main channel, and downstream in the side branch, the fluid velocity profiles are assumed to adopt those of unidirectional Poiseuille flow with prescribed flow rates. The capsule boundary is treated as a twodimensional elastic membrane developing elastic tensions and bending moments. Results are presented for different flow rates through the side branch, for different capsule sizes, and for different side branch angles. 
Brocklehurst, Paul 
Use of Weber transform and eigenfunction matching to solve hydroelastic wave interaction with a cylinder 
View Abstract
Hide Abstract

The linear twodimensional problem of hydroelastic waves diffracting around a vertical mounted cylinder is analysed. The water is of finite depth and covered by an ice sheet. The infinite ice sheet is modelled by linear elastic plate theory and is clamped to the vertical structure. A regular twodimensional incident wave approaches from infinity and is
diffracted by the cylinder. An analytic solution is found by Weber transform. We determine the ice deflection, the vertical and horizontal forces acting on the wall and the principal strains in the ice sheet. 
Eatock Taylor, Rodney 
Preliminary title "Resonances and transient responses of a floating beam" 
View Abstract
Hide Abstract

The presentation uses the integral equation for the total hydrodynamic pressure under a floating flexible beam, discretised by a Galerkin approach with Legendre polynomials as the basis functions. The link with classical ship hydroelasticity theory is reviewed, based on the modes of vibration of the "dry" structure. An eigenvalue analysis (c.f. Meylan, IWWWFB 2009) is then performed on the complete coupled hydroelastic equations which result from the Galerkin discretisation. The formulation is used with the FFT to investigate the transient interaction between a flexible beam and an incident focussed wave group (a socalled "NewWave"). 
Eloy, Christophe 
Optimisation of undulatory swimming 
View Abstract
Hide Abstract

The swimming efficiency of a rigid flapping foil has been addressed by many authors with the aim of modelling the lunate tail of fast swimmers such as sharks, dolphins and tunas. In this case, the problem is governed by few parameters: the heaving and pitching amplitudes and their phase difference. In this talk, we will first review the main results on flapping foils and then consider the general problem of a thin flexible foil undergoing an arbitrary bending motion in two dimensions. Heave and pitch motions will not be prescribed and will arise from force and torque balance on the body. We will show what are the optimal gaits in both the highspeed and lowconsumption regimes. 
Gavrilyuk, Sergey 
Solidfluid diffuse interface model 
View Abstract
Hide Abstract

We derive an Eulerian diffuse interface model for elastic solidcompressible fluid interactions in situations involving extreme deformations. Elastic effects are included following the Eulerian conservative formulation proposed in Godunov (1978), Miller and Colella (2001), Godunov and Romenski (2003), Plohr and Plohr (2005), Gavrilyuk, Favrie and Saurel (2008). We apply first the Hamilton principle of stationary action to derive the conservative part of the model. The relaxation terms are then added which are compatible with the entropy inequality. In the limit of vanishing volume fractions the Euler equations of compressible fluids and a conservative hyperelastic model are recovered. Capabilities of the model and methods are illustrated on various tests of impacts of solids moving in an ambient compressible fluid. 
Gudmestad, Ove T. 
On the concept of negative damping 
View Abstract
Hide Abstract

Damping is limiting the motions of an oscillator, a dynamic system. Different formulations for damping will be discussed. In case the forcing of the dynamic system contains terms proportional to the velocity of motion of the oscillator, the effects will contribute to the damped oscillations. Should the total damping under certain conditions become negative, the oscillations will grow until the damping again has become positive. Investigations on negative damping effects and discussions where negative damping might appear in practical applications will be given. 
Haragus, Mariana 
Transverse spectral stability of travelling periodic waves for the KP equations 
View Abstract
Hide Abstract

We discuss the question of spectral stability of onedimensional periodic travelling waves of the KadomtsevPetviashvili equations, KPI and KPII, with respect to twodimensional perturbations. We restrict to periodic waves with small amplitude. Using Floquet decomposition in the direction of propagation, and Fourier decomposition in the
perpendicular direction, the spectral stability problem is formulated in terms of the spectra of a twoparameter family of linear operators. These operators have point spectrum, only, and we show how to locate this spectrum, and in particular how to detect instabilities. 
Hirdaris, Spyros E. 
Hydroelasticity of Ships  The Classification prespective 
View Abstract
Hide Abstract

The function of Lloyd’s Register includes the setting of standards for the design, construction and maintenance of ship hulls to ensure adequate safety throughout their service life. Fundamental to this is the determination of the design loads.
The traditional design philosophy is based on closed form empirical formulae or first principles calculation procedures that, wherever applicable, assume that a ship is a rigid structure. Recently, we have experienced a stepwise increase in ship size and complexity. Examples of this include the whipping and springing response of ULCs and sloshing loads in membrane LNG ships. This raises the need to enhance our understanding on fundamental principles of fluid structure interactions with the aim to provide improved solutions to engineering problems of practical significance.
The presentation will outline some of the applications of hydroelasticity theory to ships, with particular reference to recent and ongoing developments focussing on ship design applications and the effects of nonlinearities and viscous flows. It will also discuss the longer term potential use of weakly and fully nonlinear fluidstructure interaction, as well as Navier Stokes based fluid dynamic methods, for the improved modelling of ship dynamic response problems. 
Hosking, Roger 
Theoretical Response of Floating Plates to Moving Loads 
View Abstract
Hide Abstract

In cold regions, relatively smooth floating ice sheets are often exploited by landbased vehicles and as aircraft runways. However, a moving load can produce a much larger deflexion than a stationary load, particularly near a resonance associated with waves it generates in the system. The ice sheet response has been been modelled remarkably successfully by steady state and timedependent analysis for a justifiably localised load moving over a flexible plate, with both onedimensional and twodimensional wave patterns strongly dependent on the load speed. A notable feature in the early theory was the infinite response predicted at one or more critical load speeds. It emerged that the deflexion due to a steadily moving load generally approaches a steady state, but does have a pronounced although finite peak at the critical load speed corresponding to the minimum phase speed (coincident with the group speed) of the flexuralgravity waves generated, when the plate is assumed viscoelastic (anelastic) or nonlinearity is considered. The deflexion due to a line load accelerated through this critical speed was also shown to be bounded. Nevertheless, this load speed should usually be avoided in cold region operations, unless the vehicle happens to be an icebreaking hovercraft on the St. Lawrence Seaway in Canada for example! The historical theoretical development will be reviewed, together with brief reference to relevant experimental work and also to analogous developments in rail track engineering. 
Iooss, Gérard 
Nonlinear reduction method for finding localized travelling waves in a fluid layer covered by an elastic plate 
View Abstract
Hide Abstract

This is a joint work with F. Dias. We consider a perfect fluid layer of finite depth, the flow being potential, and the fluid being covered by an elastic thin plate modeled as a Kirchhoff Love plate. Using a suitable conformal map, we formulate the problem of the search for traveling waves, as a spatial dynamical system, on which center manifold and normal form reductions methods apply. We study two typical critical situations, the first leading to 1:1 resonance (hamiltonian Hopf bifurcation), the second being a saddlecenter bifurcation. The first case may give (this depends on the depth of the fluid layer) localized solitary waves with damping oscillations at infinity, the second case gives generalized solitary waves, with (exponentially) small non damping oscillations at infinity. 
Kalisch, Henrik 
Stability of a CO2Seawater Interface 
View Abstract
Hide Abstract

Wave experiments in a trough filled with liquid CO2 are analyzed. Measurements of regular plane waves excited by a wave maker are used to determine the approximate surface tension of the interface, and the density of the liquid CO2. Thruster experiments are used to investigate the stability of this twofluid system with respect to currents near the bottom of the ocean. Conditions on the stability of these waves are given, and an approximate damping rate is found. 
Kapsenberg, Geert K. 
Slamming of ships: where are we now? 
View Abstract
Hide Abstract

The developments to date will be reviewed and a personal view on what still needs to be done will be presented. The final objective consists of two parts. Part 1 is concerned with the global response of the ship: the impulsive loads need to be predicted, the whipping response can then be calculated and as the final step the associated stresses in the structure. These stresses contribute to the fatigue damage and to the extreme values. Part 2 is concerned with the local stresses due to an impact. These stresses can result in local plastic deformations or damage to stiffeners. Hydroelastic effects are considered to be important for this case. An important aspect is to assess the space and time dependent pressure and to compare this to the “equivalent hydrostatic pressure” as is used in the rules. 
Khabakhpasheva, Tatiana 
Mathematical aspects of water wave interaction with floating elastic plate 
View Abstract
Hide Abstract

The investigation of the twodimensional problem on hydroelastic behavior of a floating plate in periodic surface waves is one of the well prevailing problems of hydroelasticity during the last fifteen years. This problem provides a basic mathematical model for describing the behaviour of very large floating structures such as floating airports. Many results on this topic were obtained numerically and do not involve mathematical substantiation of the exploitable models and methods.
As general the hydroelastic problems are coupled and can be subdivided into two or more problems of different types in mathematical sense, each of them separately being well explored (like Neumann problem for Laplace equation or initial problem for Euler beam equation). But coupled problems of hydroelasticity must be investigated and solved simultaneously, which lead to the problem of a totally new type. This indicates a hardship in this type of investigations and explains a small number of analytical results in this field.
The present paper deals with mathematical substantiation of the methods of semianalytical solution of the floating beam problem, proposed by Khabakhpasheva, Korobkin (2002). This method is based on expansion of the hydrodynamic pressure as Fourier series. After substitution of this expansion in the Euler beam equation, with corresponding edge conditions, we have the expansion of the beam deflection with respect to certain basic functions. This makes it possible to simplify the treatment of the hydrodynamic part of the problem (and solve it ones for all) and at the same time to satisfy accurately the beam boundary conditions. Finally the integral equation for the pressure leads to the infinite system of algebraic equations with respect to the principal coordinates.
It is shown, that the general matrix of this system is a symmetric matrix, which determines a HilbertSchmidt operator. This allows us to apply the method of reduction to solve this problem. It is necessary to emphasize, that in spite of a weak singularity of the pressure at the edges of the plate, elastic parameters of the plate vibration are determined with a good accuracy, because the pressure exerts only a distributed influence on the plate vibrations. 
Korobkin, Alexander 
Normal mode method in problems of liquid impact onto elastic wall 
View Abstract
Hide Abstract

Steep wave impact onto elastic wall is studied within both the incompressible liquid model and acoustic approximation. Elastic part of the wall is supported in such a way that the problems can be solved by the method of separating variables. As a result, the contribution of each elastic mode is treated separately. We study convergence of the normal mode series for the hydrodynamic pressure and the stress distribution in the wall. It is shown that the series for the pressure does not converge within the incompressible liquid model if the structural mass of the wall is neglected, and the convergence is rather slow if the structural mass is taken into account. The normal mode series derived for the distribution of the acoustic pressure along the wall well converges and the predicted pressure distributions are reliable. 
Makasyeyev, Michael 
TwoDimensional Unsteady Planing Elastic Plate 
View Abstract
Hide Abstract

The joint hydrodynamics and elasticity twodimensional problem of planing of elastic plate on the water surface is considered as boundary problem with initial conditions. For the method of fundamental solutions the joint hydroelasticity problem to the one integral equation with the unknown pressure function is reduced. The regimes of steady motion and steadystate oscillation in the issue of limiting process of time to infinity tending are obtained. Because the problem has such unknown values as wetted length, draft and trim angle, it is necessary to add the body dynamics equations to the integral equation of hydroelasticity. For the numerical solution to the integral equations the method of discrete singularities is used. The nonlinear problem of wetted length search is solved with the help of consecutive minimization procedures. 
Malenica, Sime 
Overview of hydroelastic problems in ship design 
View Abstract
Hide Abstract

An overview of the selected practical hydroelastic problems which are encountered by the ship structure in her life, are discussed. These problems are related both to the global (springing/whipping) and local (sloshing impacts, slamming,...) hydroelastic issues. Fully nonlinear problem being almost impossible to solve, within the present state of the art methods, different simplifications are proposed. The global ship hydroelasticity is solved by coupling the general 3D seakeeping code with the 3DFEM structural model, both in frequency and time domain, using the modal approach. The slamming loading is added using the so called strip approach, because the fully correct 3D slamming model is not available yet.
As far as the local hydroelastic problems are concerned, two problems are discussed namely slamming and sloshing impacts. Both are treated using the 2D semianalytical methods for fluid flow and 3D FE model for structural response. Different impact types are first identified (Wagner, Bagnold, aerated, ...) and for each of them an dedicated semianalytical fluid flow model is developed and coupled with 3D FE structural model.
Finally, the advantages and drawbacks of the pure CFD type of approaches for fluid flow are also discussed. 
Martin, Paul A. 
Multiple scattering of flexural waves by random configurations of inclusions in thin plates 
View Abstract
Hide Abstract

Flexural waves are scattered by inclusions or cavities in a thin plate. For a single inclusion, a reciprocity relation (which seems to be new) is obtained connecting coefficients in circular multipole expansions. Then, a formula for the effective wavenumber in a random arrangement of identical inclusions is derived, using the Lax quasicrystalline approximation. The formula is reminiscent of the LloydBerry formula for acoustic waves, except that it has an additional term and it involves two distinct farfield patterns.
This is joint work with William Parnell (University of Manchester). 
Metrikine, Andrei 
INTERMITTANCY OF THE SELFEXCITED VIBRATIONS OF A SUBMERGED CANTILEVER PIPE ASPIRATING WATER 
View Abstract
Hide Abstract

A concept is currently being developed by major offshore companies of liquefying natural gas offshore, on a floating barge. The equipment onboard of such a barge would need a large amount of cooling water, which will be pumped through a set of freehanging pipes. The industry is interested in the maximum flow rate that could be delivered through the pipes such that they do not become unstable due to the internal flow. Experimental and theoretical investigations have been conducted at TU Delft to answer this question. The experiments showed that a freehanging submerged pipe aspirating water at a constant speed exhibits what can be called marginal instability and intermittency at the flow velocities higher than a critical. In this talk the experimental findings will be shown and a theory will be presented that explains the intermittent behaviour. 
Meylan, Michael 
Hydroelasticity from the perspective of complex scattering frequencies 
View Abstract
Hide Abstract

The basic approach to hydroelasticity is to extend the solution for a rigid body in terms of the six rigid modes of motion, to the case of an infinite number of vibrational modes. While this is a satisfactory method to calculate the response due to an incident wave in the frequency domain, it tells us nothing about the way in which the presence of the fluid effects the vibrational modes. The concept of wet modes goes some way towards achieving this, but requires that some of the fluid effect are neglected. It we want to correctly describe the effect of the fluid on the vibrational modes we require the use of complex scattering frequencies. These are mathematically difficult objects to work with, because they require the solution be found for complex frequencies, which appear unphysical, and which cannot be found using existing commercial software. I will show how these scattering frequencies can be found by a number of methods, including an approximate method based on the solution for real frequencies. I will illustrate their importance in various ways, focusing on simple cannonical hydroelastic problems, but also showing how the method can be extended to more complicated problems, such as the vibrational response of a large container ship. 
Oliver, James 
Secondorder Wagner theory for the twodimensional water entry of a deformable body 
View Abstract
Hide Abstract

The twodimensional impact of a prescribed deformable body of small deadrise angle on a halfspace of ideal and incompressible liquid is considered within the framework of Wagner's theory from 1932. Systematic matchedasymptotic methods are used to derive expressions for the secondorder corrections to the locations of the highpressurejetroot regions and for the composite pressure and force on the body. Several applications are discussed. 
Parau, Emilian 
Threedimensional solitary waves in fluid mechanics 
View Abstract
Hide Abstract

Threedimensional fullylocalised solitarywave solutions are computed by boundary integral methods for different problems arising in fluid mechanics. In particular results for gravitycapillary waves, interfacial waves and waves under an ice sheet will be presented. Other type of threedimensional nonlinear flows will be considered. 
Peake, Nigel 
Fluidstructure interaction in the presence of mean flow 
View Abstract
Hide Abstract

The unsteady motion of an elastic plate with mean flow under conditions of heavy fluid loading is known to exhibit a range of unusual behavior, including absolute instability and negative energy waves. Here we present recent work, both numerical and asymptotic, on the relation of this infiniteplate behavior to finite systems. We consider two cases; a finite elastic plate in a rigid baffle, and a flag. The resonance of the plate is shown to be very well predicted by suitable application of infiniteplate theory, for both cases in which the infinite plate is either convectively unstable or absolutely unstable. For the flag we show that the dynamics of the free downstream end, and in particular the nature of the wake shed into the fluid, is crucial in determining the energy exchange between the plate and the flow. 
Peter, Malte 
The generalised eigenfunction method and timedependent linear waterwave impact on a vertical elastic plate 
View Abstract
Hide Abstract

The generalised eigenfunction method has recently been suggested as a powerful tool for solving timedependent linear waterwave scattering problems efficiently. Considering the problem of linear water waves impacting on a vertical elastic plate, which can be viewed as a simple model for wave interaction with elastic tank walls, we present a new elegant way of developing the generalised eigenfunction method and show how it can be used for approximating efficiently the longtime behaviour of the solution in nearresonant situations. 
Plotnikov, Pavel 
StrainGradient Theory of Hydroelastic Travelling Waves 
View Abstract
Hide Abstract

This is a mathematical study of steady twodimensional waves which propagate without changing shape on the surface of an infinitely deep fluid that moves under gravity, bounded above by a heavy, frictionless, thin (unshearable) elastic sheet. The sheet stays in contact with the zero streamline of the flow and is deformed by internal elastic forces and couples, and inertial forces and gravity, according to the laws of elasticity. The flow, which is supposed irrotational, is at rest at infinite depth and its velocity is stationary relative to a frame moving with the wave. Therefore the pressure exerted by the fluid at a steady streamline depends on its height and the fluid velocity. We study the balancing of these elastic and hydrodynamic effects to produce a steady hydroelastic wave.
To do so, it is supposed that the surface membrane is hyperelastic, with a stored energy function that depends on the stretch, the stretchgradient, and the curvature of the membrane. The problem is then formulated as one for critical points of a Lagrangian. It is notable that the stored energy for travelling waves may be nonconvex in the stretch even when the stored energy of the material at rest is convex. The existence of waves is proved by maximizing the Lagrangian in the presence of straingradient effects. An understanding of the limiting process, as the coefficient of the straingradient term in the elastic energy tends to zero, is the main purpose of the investigation. 
Schulkes, Ruben 
Some fluidstructure interaction problems in the energy industry 
View Abstract
Hide Abstract

Interactions between fluids and flexible structures are of occur in many different areas in the energy industry. In this presentation we look at a number of actual specific fluidstructure interactions problems.
The problem of ballooning methanol injection lines is presented in some detail. The dynamic response of these injections lines and typical relaxation time scales are considered.
Some challenges related to fluidstructure interactions related to renewable energy are also presented. 
Smith, Frank T. 
Manybody problems in fluids 
View Abstract
Hide Abstract

The presentation will be based on three main aspects of nonlinear fluidbody or fluidbodies interaction and will mostly address the regime of relatively high flow rates. First will be a discussion of various mathematical problems that have arisen in recent times concerned with multibody interactions in surrounding fluid and the effects of increases in the body count, of different geometries and of different flow rates for example. Second, a particular problem will be described which involves one or many moving rigid bodies whose individual and combined motion affects and is affected by all the surrounding fluid regions. Here the emphasis is on the initial value context and the system stability, the eventual occurrence of clashes between the bodies and the behaviour of the system when the number of bodies is large. The third aspect will be concerned with the implications for one or many moving bodies but of flexible shape(s), within the surrounding fluid. All three have been sparked off by biomedical or industrial modelling interests. 
Squire, Vernon 
Hydroelastic Chronicles of a Gelid Mathematician 
View Abstract
Hide Abstract

The field of ocean wave / sea ice interactions and the complementary topic of the hydroelastic analysis of pontoontype, very large floating structures have seen a massive resurgence of interest in the last two decades, documented in syntheses by Squire et al. (1995) and Squire (2007, 2008). Many physical situations that remained unresolved in the early nineties have now been comprehensively studied by mathematicians and engineers using sophisticated techniques that have allowed accurate models to be created that replicate natural wave phenomena occurring in the polar and subpolar seas. Associated with this has been a greater appreciation of the need to deal with heterogeneity, as the material properties and topography of all types of ice in the ocean, i.e. icebergs and ice islands as well as sea ice, varies spatially and temporally. Unfortunately, although models have become ever more sophisticated, a commensurate evolvement of experimental work, either in the laboratory or in the field, has not kept pace, so the data available for testing theory are still much the same as those reported in 1995. The effect of global climate warming is particularly pertinent, as ocean waves are likely to be more intense, to penetrate further and to carry a more destructive payload in a degraded sea ice lamella. I will chronicle the research, paying particular attention to the most recent developments and to where the next few years may take us.
Squire, V.A., Dugan, J.P., Wadhams, P., Rottier, P.J. and Liu, A.K. Of ocean waves and sea ice. Annual Review of Fluid Mechanics 27: 115–168 (1995);
Squire, V.A. Of ocean waves and seaice revisited. Cold Regions Science & Technology 49(2): 110–133 (2007);
Squire, V.A. Synergies between VLFS hydroelasticity and seaice research. International Journal of Offshore and Polar Engineering 18(4): 241–253 (2008). 
Tassin, Alan 
Threedimensional water impact problems: numerical modelling based on the Wagner theory and experiments. 
View Abstract
Hide Abstract

Some bulbous bows, also used as sonar domes, are made of fibre reinforced plastic and experience high transient loading during slamming events. It is therefore important to study the dynamic response of the structure to water impacts. A preliminary requirement for that is to accurately predict the transient pressure distribution acting on a complex threedimensional body during water entry. Although CFD techniques make it possible to accurately describe this phenomenon, shipbuilders still need simple practical and efficient tools for preliminary design or optimization. Several models of water impact based on the Wagner approach have been developed for this purpose but most of them are restricted to the twodimensional case. As the strip method is inaccurate for nonslender bodies, a numerical method has been developed to solve the threedimensional Wagner problem. The predictions of slamming loads obtained with the proposed model have been compared to results of CFD models and to experimental data. The proposed method makes it possible to quickly compute the pressure distribution occurring during slamming events and will be used to analyse the dynamic response of bulbous bows to such loads. 
Tucsnak, Marius 
Mathematical analysis of some swimming mechanisms 
View Abstract
Hide Abstract

We are interested in the mathematical analysis of selfpropelles motions of solids in a fluid (swimming). The high and the low Reynolds regimes are considered, the latter from a control theoretic view point. 
VandenBroeck, JeanMarc 
Nonlinear waves under a sheet of ice. 
View Abstract
Hide Abstract

We consider the propagation of nonlinear waves in a layer of water bounded above by a sheet of ice. Weakly nonlinear and fully nonlinear solutions are considered. Periodic waves and solitary waves with decaying oscillatory tails are presented. 